Extravasation of drug delivery nano-particles into tumors and applications in cancer treatment
In many tumors, small pores, several hundred nanometers in size, develop through blood vessel
walls as a result of abnormal endothelial cell growth near the tumor. Drug delivery nanoparticles
may extravasate through these pores to the tumors, enabling payload delivery to the tumor itself.
This rate of extravasation is very important in the efficacy of a drug treatment, and it depends
on flow conditions such as shear rate and particle size. Experimental evidence points towards
a strong dependence on particle shape as well (Smith et al. Nano Lett., 2012, 12(7), pp 3369-
3377). As a result, nanoparticles can be engineered with a particular geometry to preferentially
target a tumor, dependent on the flow conditions, leading to better targeting and improved efficacy of treatment.

Fig 1: Schematic of extravasation process and length scales involved. Nps are bumped to the periphery of the channel where they encounter a linear flow and extravasate through endothelial cell layer pores.

Motivated by these recent studies,
We developed a singular perturbation theory and Brownian dynamics simulations to quantify
the extravasation rate of Brownian particles in a shear flow over a circular pore with a local
lumped mass transfer resistance. The analytic theory is an expansion of Sherwood
number in the limit of vanishing Peclet number, which is the ratio of convective fluxes to
diffusive fluxes on the length scale of the pore. This model is useful in analogs in electrochemistry and heat transfer. Our work provides theory to calculate Sherwood number for the case of finite mass transfer resistance
at small Peclet numbers, which are physiologically important in extravasation of finite-sized particles.
The theory and simulation for spherical nps agree well with each other across all range of Damkohler numbers and short range of Peclet numbers.

Fig 2: Sherwood number plotted against Damkohler number for Peclet number of 0.4. Theoretical and simulation values show good agreement. Also, physiologically, Pe ~ O(1)

Our simulation now can also incorporate the effect of size and shape in the presence or absence of a suction flow.
It is observed that large particles in general find it more difficult to go through the pore due to excluded volume effects. Rod-shaped particles find it easier to extravasate than their similar sized spherical counterparts with or without suction flow due to excluded volume. In the presence of a pressure gradient driven suction flow, there is little distinction
between the extravasation rates of large and small rod-shaped particles indicating that
suction dominates the physics of the problem.